The typical positions of (60 initial and 60 sequential) samples in Seq-VHGPR are shown in figure REF (a)(b), as well as the the learned functions \(f(x_1,x_2)\)  and \(\gamma (x_1,x_2)\) . Similar to the 1D case, the sequential samples are expected to concentrate in regions where \(\mathbb {P}(S(x_1,x_2, \omega )>\delta ) p_{X_1,X_2}(x_1,x_2)\)  is maximized, i.e., the four regions enclosed by -4 contour lines in figure REF (c). As shown in figure REF (a)(b), most sequential samples lie in three out of the four regions (although the situation depends on the initial samples and for some cases all four regions can be filled). The difficulty of the sequential samples transiting to all four regions within 60 samples can be anticipated, which is consistent with the observation in the case of deterministic ItR if the U criterion is used as the acquisition function [1]}. While the design of better acquisition function is possible referring to the counterpart in the deterministic case [2]}, the current results already show the adequacy of Seq-VHGPR in estimating the exceeding probability (even if not all four regions are filled and the estimation of \(\gamma (x_1,x_2)\)  is relatively less accurate than that of \(f(x_1,x_2)\) ).
